Distribution of prime numbers modulo $4$

In general, if $\gcd(a,b) = 1$, the number of primes which are of the form $b$ modulo $a$ is asymptotic to $\dfrac{\pi(x)}{\varphi(a)}$ where $\pi(x)$ is the number of primes $\le x$. As you see the asymptotic formula is independent of $b$ hence for a given modulo all residues occur equally in the long run.

Despite of Chebychev's bias (more primes with residue $3$ occur usually in practice), in the long run, the ratio is $1:1$.